The Spectral Ewald method for singly periodic domains

It is well-known that by placing judiciously chosen image point forces and doublets to the Stokeslet above a flat wall, the no-slip boundary condition can be conveniently imposed on the wall [Blake, J. R. Math. Proc. Camb. Philos. Soc. 70(2), 1971: 303.]. However, to further impose periodic boundary conditions on directions parallel to the wall usually involves tedious derivations because single or double periodicity in Stokes flow may require the periodic unit to have no net force, which is not satisfied by the well-known image system. In this work we present a force-neutral image system. This neutrality allows us to represent the Stokes image system in a universal formulation for non-periodic, singly periodic and doubly periodic geometries. This formulation enables the black-box style usage of fast kernel summation methods. We demonstrate the efficiency and accuracy of this new image method with the periodic kernel independent fast multipole method in both non-periodic and doubly periodic geometries. We then extend this new image system to other widely used Stokes fundamental solutions, including the Laplacian of the Stokeslet and the Rotne-Prager-Yamakawa tensor.

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“…The final combination stages of Eq. (15) and (16) have negligible cost, similar to the case for the Stokeslet shown in Table 1.…”

Section: The Kernel Sums For Polydisperse Systemssupporting

confidence: 55%

Universal image systems for non-periodic and periodic Stokes flows above a no-slip wall

Yan

Shelley

2018

Journal of Computational Physics

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“…The corresponding expressions for the stresslet and rotlet are tabulated for reference in the appendix in equations (14) and (15).…”

Section: The Ewald Decompositionmentioning

confidence: 99%

“…Otherwise, for every direction that is not periodic, oversampling of the FFTs becomes necessary to compute the aperiodic convolution, and this increases the computational cost. The work in [13] illustrates the use of the SE method for a 2-periodic sum of stokeslets, while in [14] the SE method is adapted for 1-periodic sums in the context of electrostatics. The case of free-space sum of stokeslets (no periodicity) is the most challenging for the SE method and it was solved recently by Klinteberg et al [2] by combining two different ideas.…”

Section: Introductionmentioning

confidence: 99%

Fast Ewald summation for Green’s functions of Stokes flow in a half-space

Srinivasan

Tornberg

2018

Res Math Sci

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Recently, Gimbutas et al in [1] derived an elegant representation for the Green's functions of Stokes flow in a half-space. We present a fast summation method for sums involving these halfspace Green's functions (stokeslets, stresslets and rotlets) that consolidates and builds on the work by Klinteberg et al [2] for the corresponding free-space Green's functions. The fast method is based on two main ingredients: The Ewald decomposition and subsequent use of FFTs. The Ewald decomposition recasts the sum into a sum of two exponentially decaying series: one in real-space (short-range interactions) and one in Fourier-space (long-range interactions) with the convergence of each series controlled by a common parameter. The evaluation of short-range interactions is accelerated by restricting computations to neighbours within a specified distance, while the use of FFTs accelerates the computations in Fourier-space thus accelerating the overall sum. We demonstrate that while the method incurs extra costs for the half-space in comparison to the freespace evaluation, greater computational savings is also achieved when compared to their respective direct sums. 0 conditionally convergent sum into two rapidly converging sums -one in real space and one in Fourier space. The computational complexity is however quadratic in the number of points. The survey by Deserno and Holm [7] traces the development of fast methods based on FFTs for acceleration of the Fourier space sum in the context of electrostatics and molecular dynamics. An early method called Smooth Particle Mesh Ewald (SPME) [8] was utilized by Saintillan et al [9] for fast evaluation of periodic stokeslet sums. Later, Tornberg et al developed a Spectral Ewald (SE) method for the 3periodic sum of stokeslets [10], stresslets [11] and rotlets [12] that is spectrally accurate and recovers the exponentially fast convergence of the Ewald sums that traditional Particle Mesh Ewald approaches cannot.The SE method is best suited for the 3-periodic case. Otherwise, for every direction that is not periodic, oversampling of the FFTs becomes necessary to compute the aperiodic convolution, and this increases the computational cost. The work in [13] illustrates the use of the SE method for a 2-periodic sum of stokeslets, while in [14] the SE method is adapted for 1-periodic sums in the context of electrostatics. The case of free-space sum of stokeslets (no periodicity) is the most challenging for the SE method and it was solved recently by Klinteberg et al [2] by combining two different ideas. The first idea is the free-space solution of harmonic and biharmonic equations using FFT on a uniform grid by Vico et al [15] that amounts to the convolution of harmonic/biharmonic (radial) kernels with source terms by FFT on a uniform grid, and the second idea is that the stokeslet, stresslet and rotlet kernels, though not radial, can be expressed as a linear combination of differential operations on the harmonic or biharmonic kernels.A popular method ideally suited for free-space problems is t...

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“…Analytical formulas useful for validation are derived in the singly and doubly periodic cases, completing the formulas previously derived by [20] for the doubly periodic stokeslet. The SE method presented in this paper furthermore uses the polynomial Kaiser-Bessel (PKB) window introduced by [51] and [24], and the adaptive Fourier transform (AFT) introduced by [52] for the singly and doubly periodic cases.…”

Section: Introductionmentioning

confidence: 99%

Fast Ewald summation for Stokes flow with arbitrary periodicity

Bagge¹,

Tornberg²

2022

Preprint

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A fast and spectrally accurate Ewald summation method for the evaluation of stokeslet, stresslet and rotlet potentials of three-dimensional Stokes flow is presented. This work extends the previously developed Spectral Ewald method for Stokes flow to periodic boundary conditions in any number (three, two, one, or none) of the spatial directions, in a unified framework. The periodic potential is split into a short-range and a long-range part, where the latter is treated in Fourier space using the fast Fourier transform. A crucial component of the method is the modified kernels used to treat singular integration. We derive new modified kernels, and new improved truncation error estimates for the stokeslet and stresslet. An automated procedure for selecting parameters based on a given error tolerance is designed and tested. Analytical formulas for validation in the doubly and singly periodic cases are presented. We show that the computational time of the method scales like O(N log N ) for N sources and targets, and investigate how the time depends on the error tolerance and window function, i.e. the function used to smoothly spread irregular point data to a uniform grid. The method is fastest in the fully periodic case, while the run time in the free-space case is around three times as large. Furthermore, the highest efficiency is reached when applying the method to a uniform source distribution in a primary cell with low aspect ratio. The work presented in this paper enables efficient and accurate simulations of three-dimensional Stokes flow with arbitrary periodicity using e.g. boundary integral and potential methods.

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The Spectral Ewald method for singly periodic domains

Cited by 14 publications

References 30 publications

Universal image systems for non-periodic and periodic Stokes flows above a no-slip wall

Universal image systems for non-periodic and periodic Stokes flows above a no-slip wall

Fast Ewald summation for Green’s functions of Stokes flow in a half-space

Fast Ewald summation for Stokes flow with arbitrary periodicity

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